Since no one else said it, the short-cut to compile (build) a C# app in Visual Studio Code (VSCode) is SHIFT+CTRL+B.

If you want to see the build errors (because they don‘t pop-up by default), the shortcut is SHIFT+CTRL+M.

(I know this question was asking for more than just the build shortcut. But I wanted to answer the question in the title, which wasn‘t directly answered by other answers/comments.)